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Need help for one-to one linear transformation

  1. May 21, 2006 #1
    Question: Let T:V-->W and S:W-->U be linear transformation.Show that
    1) If T and S are one-to-one,then ST is one-to one
    2) If ST is one-to-one,then T is one-to-one
    3)Give example of two linear transformations T and S, such
    that ST is one-to-one ,but S is not.

    For 1),Since T is one-to-one,there is some vector in V,such that T(v)=O
    Since S is one-to-one,there is some vector in W,such that S(w)=O
    If ST is one-to-one,then there is some vector t in T,such that
    ST(t)=S(T(t))=O,so how could I proof T(t)=w?

    How about question 2) &3)
  2. jcsd
  3. May 21, 2006 #2


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    How many are there?
  4. May 21, 2006 #3
    It could be more than One vector in V, such that T(v)=O,I know Ker T=O

  5. May 21, 2006 #4
    Sorry, I made a mistake,It has only One vector in V, such that T(v)=O

  6. May 22, 2006 #5
    Here is one way to do:
    2) If ST is one-to-one,then T is one-to-one

    Proof: Suppose T is not one-to-one. Then there exists x, y in V, x != y, such that T(x) = T(y).Then ST(x) = S(T(x)) = S(T(y)) = ST(y) but x != y, and thus ST is not one-to-one.

    You should try to give a direct proof for practice. Also notice the fact that S and T are linear transformations was never used and so this result holds for functions in general.
    Last edited: May 22, 2006
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